how to simplify (x 3-x)/(x-1)?

To simplify the expression (x^3 - x)/(x - 1), we can use polynomial division or factorization.

Method 1: Polynomial Division
1. Write the expression as a long division problem, with (x^3 - x) as the dividend and (x - 1) as the divisor.
2. Divide the first term of the dividend (x^3) by the first term of the divisor (x). The result is x^2.
3. Multiply (x - 1) by x^2 and subtract the result (x^2 - x^2) from the dividend.
4. Bring down the next term of the dividend, which is 0x^2 (because we subtracted x^2 from x^2 and got 0).
5. Repeat the steps until you reach the last term of the dividend.
6. The final result of the division will be the quotient.

Method 2: Factorization
1. Factor the numerator, (x^3 - x), and the denominator, (x - 1), as much as possible.
- Factoring the numerator: x(x^2 - 1) = x(x + 1)(x - 1)
- Factoring the denominator: x - 1
2. Cancel out the common factors between the numerator and the denominator.
- Common factor: x - 1
3. The simplified form is the remaining factors in the numerator divided by the remaining factors in the denominator.

Therefore, the simplified form of (x^3 - x)/(x - 1) is x(x + 1).